A companion to S.Lang's Algebra 4ed. by Bergman G.M. PDF

By Bergman G.M.

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Extra info for A companion to S.Lang's Algebra 4ed.

Example text

Next, an unfamiliar, but nevertheless similar situation: Suppose we consider two mathematical objects X0 and X1 , and the set S of all maps among these. This set falls naturally into four pieces: S00 , consisting of all maps from X0 to itself, S01 , consisting of maps from X0 to X1 , etc.. One can compose various of these maps, but not every map is composable with every other, so one does not have a composition operation S × S → S, but rather, a family of 8 operations, Sjk × Sij → Sik (i, j, k ∈{0,1}).

Similarly, the concept of a group is motivated by considering the properties of the set of automorphisms of a mathematical object. The set of such automorphisms has a structure of monoid, but is also closed under taking inverses; this gives another operation, which satisfies familiar identities. G. M. 34 If one considers the set of subobjects of a mathematical object, there are no operations one can automatically define on this set, but there is a relation of ‘‘inclusion’’. A generalization and abstraction of the properties of this relation motivates the definition of partially ordered set.

We have also noted that developments differ in whether they assume the axiom CAT 1 (morphism-sets are disjoint). In the above matters I shall follow the choices Lang has made. On the other hand, as mentioned earlier, I will not follow Lang where he has introduced various pieces of very nonstandard terminology and notation; thus, what he calls ‘‘universal repelling’’ and ‘‘universal attracting’’ objects of a category we shall call ‘‘initial’’ and ‘‘terminal’’ objects, what he calls ‘‘stripping’’ functors we shall call ‘‘forgetful G.

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A companion to S.Lang's Algebra 4ed. by Bergman G.M.


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